code: def group_isomorphism(G, H, isomorphism=True):
      """
      Compute an isomorphism between 2 given groups.
  
      Parameters
      ==========
  
      G : A finite ``FpGroup`` or a ``PermutationGroup``.
          First group.
  
      H : A finite ``FpGroup`` or a ``PermutationGroup``
          Second group.
  
      isomorphism : bool
          This is used to avoid the computation of homomorphism
          when the user only wants to check if there exists
          an isomorphism between the groups.
  
      Returns
      =======
  
      If isomorphism = False -- Returns a boolean.
      If isomorphism = True  -- Returns a boolean and an isomorphism between `G` and `H`.
  
      Examples
      ========
  
      >>> from sympy.combinatorics import free_group, Permutation
      >>> from sympy.combinatorics.perm_groups import PermutationGroup
      >>> from sympy.combinatorics.fp_groups import FpGroup
      >>> from sympy.combinatorics.homomorphisms import group_isomorphism
      >>> from sympy.combinatorics.named_groups import DihedralGroup, AlternatingGroup
  
      >>> D = DihedralGroup(8)
      >>> p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
      >>> P = PermutationGroup(p)
      >>> group_isomorphism(D, P)
      (False, None)
  
      >>> F, a, b = free_group("a, b")
      >>> G = FpGroup(F, [a**3, b**3, (a*b)**2])
      >>> H = AlternatingGroup(4)
      >>> (check, T) = group_isomorphism(G, H)
      >>> check
      True
      >>> T(b*a*b**-1*a**-1*b**-1)
      (0 2 3)
  
      Notes
      =====
  
      Uses the approach suggested by Robert Tarjan to compute the isomorphism between two groups.
      First, the generators of ``G`` are mapped to the elements of ``H`` and
      we check if the mapping induces an isomorphism.
  
      """
      if not isinstance(G, (PermutationGroup, FpGroup)):
          raise TypeError(
              "The group must be a PermutationGroup or an FpGroup"
          )
      if not isinstance(H, (PermutationGroup, FpGroup)):
          raise TypeError(
              "The group must be a PermutationGroup or an FpGroup"
          )
  
      if isinstance(G, FpGroup) and isinstance(H, FpGroup):
          G = simplify_presentation(G)
          H = simplify_presentation(H)
          # Two infinite FpGroups with the same generators are isomorphic
          # when the relators are same but are ordered differently.
          if (
              G.generators == H.generators
              and (G.relators).sort() == (H.relators).sort()
          ):
              if not isomorphism:
                  return True
              return (
                  True,
                  homomorphism(G, H, G.generators, H.generators),
              )
  
      #  `_H` is the permutation group isomorphic to `H`.
      _H = H
      g_order = G.order()
      h_order = H.order()
  
      if g_order is S.Infinity:
          raise NotImplementedError(
              "Isomorphism methods are not implemented for infinite groups."
          )
  
      if isinstance(H, FpGroup):
          if h_order is S.Infinity:
              raise NotImplementedError(
                  "Isomorphism methods are not implemented for infinite groups."
              )
          _H, h_isomorphism = H._to_perm_group()
  
      if (g_order != h_order) or (G.is_abelian != H.is_abelian):
          if not isomorphism:
              return False
          return (False, None)
  
      if not isomorphism:
          # Two groups of the same cyclic numbered order
          # are isomorphic to each other.
          n = g_order
          if (igcd(n, totient(n))) == 1:
              return True
  
      # Match the generators of `G` with subsets of `_H`
      gens = list(G.generators)
      for subset in itertools.permutations(_H, len(gens)):
          images = list(subset)
          images.extend(
              [_H.identity] * (len(G.generators) - len(images))
          )
          _images = dict(zip(gens, images))
          if _check_homomorphism(G, _H, _images):
              if isinstance(H, FpGroup):
                  images = h_isomorphism.invert(images)
              T = homomorphism(G, H, G.generators, images, check=False)
              if T.is_isomorphism():
                  # It is a valid isomorphism
                  if not isomorphism:
                      return True
                  return (True, T)
  
      if not isomorphism:
          return False
      return (False, None)
  
imports: import itertools
  from sympy.combinatorics.fp_groups import FpGroup, FpSubgroup, simplify_presentation
  from sympy.combinatorics.free_groups import FreeGroup
  from sympy.combinatorics.perm_groups import PermutationGroup
  from sympy.core.intfunc import igcd
  from sympy.ntheory.factor_ import totient
  from sympy.core.singleton import S
label_desc: Use image node_id 5 for calling a global function with example usage: group_isomorphism(G, H, isomorphism) and returns: False,  None, False,  None, False, True,  homomorphism, False, True, True, True,  T, True